Question: Determine how many solutions exist for the system of equations. ${6x-3y = -3}$ ${3x-3y = -18}$
Convert both equations to slope-intercept form: ${6x-3y = -3}$ $6x{-6x} - 3y = -3{-6x}$ $-3y = -3-6x$ $y = 1+2x$ ${y = 2x+1}$ ${3x-3y = -18}$ $3x{-3x} - 3y = -18{-3x}$ $-3y = -18-3x$ $y = 6+x$ ${y = x+6}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 2x+1}$ ${y = x+6}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.